Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

Since any non-trivial group always has two trivial subgroups (the trivial one and the whole group), we're looking for a group with one single non-trivial subgroup...hint: how many primes can divide the group's order?

No - proper means it is properly contained in it. That is to say, \(\displaystyle H \lneq G\). So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).

As a warning to the OP, this does not classify them all. Another example would be \(\displaystyle V_4\), the klein 4-group (or, more generally, the cross-product of two groups of prime order).

No - proper means it is properly contained in it. That is to say, \(\displaystyle H \lneq G\). So we are looking for a group with precisely two non-trivial subgroups (although I think your hint is still the way to go).

Yes. I oversaw the word "proper" in the OP, but still the hint remains...though nevertheless there are OTHER examples: for example, a cyclic group of order 10...